Whether a group $G$ is cyclic or not.

Question

Let $G$ be a group. If for any two subgroups $A,B$ in $G$, either $A\subset B $ or $B\subset A$ , then $G$ is cyclic.

This asked in a exam, but I give the answer as no, as because in cyclic group , there is only unique subgroup of a particular order.

But, what is the actual conclusion and am I right?

Answer 1

If $G$ is finite the result is true. Let $x$ with the maximal order, for every $y\in G$, the group generated by $y$ $gr(y)\subset gr(x)$ implies that $G=gr(x)$.

Answer 2

I don't understand what you mean by "but I give the answer as no, as because in cyclic group , there is only unique subgroup of a particular order."

The claim is true, and I would prove it by contradiction. Suppose $G$ is not cyclic. So a minimal collection of generators has at least two generators; pick two. What can you say about the two subgroups generated by these? Is either subgroup contained within the other?